If $\left\{ \left. {{x}_{1}},{{x}_{2}},...,{{x}_{n}} \right\} \right.$and $\left\{ \left. {{y}_{1}},{{y}_{2}},...,{{y}_{n}} \right\} \right.$ are basis of a vector space V ,and for every $j=1,2,...n$
$B{{x}_{j}}=\sum\limits_{i=i}^{n}{{{a}_{ij}}{{x}_{i}}}$ and $C{{y}_{j}}=\sum\limits_{i=i}^{n}{{{a}_{ij}}{{y}_{i}}}$. What is the relation between the linear transformation B and C ?
my intuition is that they are similar .I tried to subtrabct B and C .Thats so far
If $\varphi: V\to V$ is the linear transformation defined by $\varphi(x_i)=y_i$, then \begin{equation} \varphi^{-1}\circ C\circ\varphi(x_j) = \varphi^{-1}\circ C y_j = \varphi^{-1}\left(\sum_j a_{i j} y_i\right) = \sum_j a_{i j} x_i = B x_j \end{equation} Hence \begin{equation} B = \varphi^{-1}\circ C\circ\varphi \end{equation}